OBLONG NUMBERS 
83 
‘ unlimited ’ 1 (cf. the Pythagorean scheme of ten pairs of 
opposites, where odd, limit and square in one set are opposed 
to even, unlimited and oblong respectively in the other). 2 For, 
while the adding of the successive odd numbers as gnomons 
round 1 gives only one form, the square, the addition of the 
successive even numbers to 2 gives a succession of £ oblong ’ 
numbers all dissimilar in form, that is to say, an infinity of 
forms. This seems to be indicated in the passage of Aristotle’s 
Physics where, as an illustration of the view that the even 
is unlimited, he says that, where gnomons are put round 1, 
the resulting figures are in one case always different in 
species, while in the other they always preserve on# form 3 ; 
the one form is of course the square formed by adding the 
odd numbers as gnomons round 1; the words kccl yco/ny 
(‘ and in the separate case ’, as we may perhaps translate) 
imperfectly describe the second case, since in that case 
even numbers are put round 2, not 1, but the meaning 
seems clear. 4 It is to be noted that the word eVe/oop)*??? 
(‘ oblong’) is in Theon of Smyrna and Nicomachus limited to 
numbers which are the product of two factors differing by 
unity, while they apply the term TrpoprjKys ('prolate’, as it 
were) to numbers which are the product of factors differing 
by two or more (Theon makes TrpoprjKijs include eTepoprjKys). 
In Plato and Aristotle irepopyKijs has the wider sense of any 
non-square number with two unequal factors. 
It is obvious that any ‘oblong’ number n{n+ 1) is the 
sum of two equal triangular numbers. Scarcely less obvious 
is the theorem of Theon that any square number is made up 
of two triangular numbers 5 ; in this case, as is seen from the 
1 Arist. Metaph. A. 5, 986 a 17. 
2 lb. A. 5, 986 a 23-26. 
3 Arist. Phys. iii. 4, 208 a 10-15. 
4 Cf. Plut. (?) Stob. Ed. i. pr. 10, p. 22. 16 Wachsmuth. 
5 Theon of Smyrna, p. 41. 8-8. 
O 2